from builtins import range
import numpy as np

# https://github.com/cs231n/cs231n.github.io/tree/master/assignments/2019

def affine_forward(x, w, b):
    """
    Computes the forward pass for an affine (fully-connected) layer.

    The input x has shape (N, d_1, ..., d_k) and contains a minibatch of N
    examples, where each example x[i] has shape (d_1, ..., d_k). We will
    reshape each input into a vector of dimension D = d_1 * ... * d_k, and
    then transform it to an output vector of dimension M.
    输入x的形状为(N, d_1, ..., d_k)，其中包含了N个示例组成的minibatch，
    每个示例x[i]都有形状(d_1, ..., d_k)。
    我们将把每个输入重新塑成一个维向量D = d_1 * ... * d_k，然后将其转换为M维的输出向量。
    
    Inputs:
    - x: A numpy array containing input data, of shape (N, d_1, ..., d_k)
    - w: A numpy array of weights, of shape (D, M)
    - b: A numpy array of biases, of shape (M,)

    Returns a tuple of:
    - out: output, of shape (N, M)
    - cache: (x, w, b)
    """
    out = None
    ###########################################################################
    # TODO: Implement the affine forward pass. Store the result in out. You   #
    # will need to reshape the input into rows.                               #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 执行仿射前向传播。将结果存储在out中。您需要将输入reshape成行。
    pass
    
    x_reshape = x.reshape((x.shape[0], -1))
    out = x_reshape.dot(w) + b
    
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    cache = (x, w, b)
    return out, cache


def affine_backward(dout, cache):
    """
    Computes the backward pass for an affine layer.

    Inputs:
    - dout: Upstream derivative, of shape (N, M)
    - cache: Tuple of:
      - x: Input data, of shape (N, d_1, ... d_k)
      - w: Weights, of shape (D, M)
      - b: Biases, of shape (M,)

    Returns a tuple of:
    - dx: Gradient with respect to x, of shape (N, d1, ..., d_k)
    - dw: Gradient with respect to w, of shape (D, M)
    - db: Gradient with respect to b, of shape (M,)
    """
    x, w, b = cache
    dx, dw, db = None, None, None
    ###########################################################################
    # TODO: Implement the affine backward pass.                               #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass 
#     '''
    x_reshape = x.reshape(x.shape[0] , -1)  
    dx = dout.dot(w.T)
    dx = dx.reshape(*x.shape)
    dw = x_reshape.T.dot(dout)
    db = np.sum(dout, axis = 0)
#     '''
    '''
    dx = np.dot(dout,w.T)
    dx = np.reshape(dx,x.shape)
    dw = np.dot(np.reshape(x, (x.shape[0], -1)).T,dout)
    db = np.sum(dout,axis=0)
    '''
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx, dw, db


def relu_forward(x):
    """
    Computes the forward pass for a layer of rectified linear units (ReLUs).

    Input:
    - x: Inputs, of any shape

    Returns a tuple of:
    - out: Output, of the same shape as x
    - cache: x
    """
    out = None
    ###########################################################################
    # TODO: Implement the ReLU forward pass.                                  #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass
#     out = x * (x >= 0)
    out = np.maximum(x,0)

    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    cache = x
    return out, cache


def relu_backward(dout, cache):
    """
    Computes the backward pass for a layer of rectified linear units (ReLUs).

    Input:
    - dout: Upstream derivatives, of any shape
    - cache: Input x, of same shape as dout

    Returns:
    - dx: Gradient with respect to x
    """
    dx, x = None, cache
    ###########################################################################
    # TODO: Implement the ReLU backward pass.                                 #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass
    dx = (x >= 0) * dout
    '''
    dx=dout
    dx[x<=0]=0
    '''
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx


def batchnorm_forward(x, gamma, beta, bn_param):
    """
    Forward pass for batch normalization.

    During training the sample mean and (uncorrected) sample variance are
    computed from minibatch statistics and used to normalize the incoming data.
    During training we also keep an exponentially decaying running mean of the
    mean and variance of each feature, and these averages are used to normalize
    data at test-time.
    在训练中，样本均值和(uncorrected)样本方差由minibatch统计数据计算出来，并用于normalize输入数据。
    在训练中，我们还保持每个特征的均值和方差的指数衰减的运行平均值，这些平均值用于在测试时normalize数据。
    At each timestep we update the running averages for mean and variance using
    an exponential decay based on the momentum parameter:
    在每个timestep，我们更新的平均和方差的运行平均值，使用基于动量参数的指数衰减:
    running_mean = momentum * running_mean + (1 - momentum) * sample_mean
    running_var = momentum * running_var + (1 - momentum) * sample_var

    Note that the batch normalization paper suggests a different test-time
    behavior: they compute sample mean and variance for each feature using a
    large number of training images rather than using a running average. For
    this implementation we have chosen to use running averages instead since
    they do not require an additional estimation step; the torch7
    implementation of batch normalization also uses running averages.
    注意，batch normalization的论文提出了一种不同的test-time行为:
    他们计算每个特征的样本均值和方差，使用大量的训练图像，而不是使用运行平均值。
    对于这个实现，我们选择使用运行平均值，因为它们不需要额外的估计步骤;
    batch normalization的torch7实现也使用运行平均值。
    
    Input:
    - x: Data of shape (N, D)
    - gamma: Scale parameter of shape (D,)
    - beta: Shift paremeter of shape (D,)
    - bn_param: Dictionary with the following keys:
      - mode: 'train' or 'test'; required
      - eps: Constant for numeric stability
      - momentum: Constant for running mean / variance.
      - running_mean: Array of shape (D,) giving running mean of features
      - running_var Array of shape (D,) giving running variance of features

    Returns a tuple of:
    - out: of shape (N, D)
    - cache: A tuple of values needed in the backward pass
    """
    mode = bn_param['mode']
    eps = bn_param.get('eps', 1e-5)
    momentum = bn_param.get('momentum', 0.9)

    N, D = x.shape
    running_mean = bn_param.get('running_mean', np.zeros(D, dtype=x.dtype))
    running_var = bn_param.get('running_var', np.zeros(D, dtype=x.dtype))

    out, cache = None, None
    if mode == 'train':
        #######################################################################
        # TODO: Implement the training-time forward pass for batch norm.      #
        # Use minibatch statistics to compute the mean and variance, use      #
        # these statistics to normalize the incoming data, and scale and      #
        # shift the normalized data using gamma and beta.                     #
        #                                                                     #
        # You should store the output in the variable out. Any intermediates  #
        # that you need for the backward pass should be stored in the cache   #
        # variable.                                                           #
        #                                                                     #
        # You should also use your computed sample mean and variance together #
        # with the momentum variable to update the running mean and running   #
        # variance, storing your result in the running_mean and running_var   #
        # variables.                                                          #
        #                                                                     #
        # Note that though you should be keeping track of the running         #
        # variance, you should normalize the data based on the standard       #
        # deviation (square root of variance) instead!                        # 
        # Referencing the original paper (https://arxiv.org/abs/1502.03167)   #
        # might prove to be helpful.                                          #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        # 对 batch norm 完成 training-time 前向传播。
#         https://zhuanlan.zhihu.com/p/21560667
        pass
        # Compute output
        # 使用minibatch统计数据来计算平均值和方差，
        # 使用这些统计数据来normalize输入数据，
        mu = x.mean(axis=0)
        xc = x - mu
        var = np.mean(xc ** 2, axis=0)
        std = np.sqrt(var + eps)
        xn = xc / std
        # 使用gamma和beta来缩放和移动normalized数据。
        # 您应该将输出存储在变量out中。
        out = gamma * xn + beta
        # 用于反向传播的任何intermediates都应该存储在cache变量中。
#         cache = (mode, x, gamma, xc, std, xn, out) # 官方答案，见assignment3
        cache = (mode, x, gamma, mu, var, eps, xn)
        # 您还应该使用计算出的样本均值和方差以及动量变量来更新运行均值和运行方差，
        # 将结果存储在running_mean和running_var变量中。
        # 请注意，虽然您应该跟踪运行的方差，但您应该根据标准偏差(方差的平方根)normalize数据!
        # Update running average of mean
        running_mean *= momentum
        running_mean += (1 - momentum) * mu
        # Update running average of variance
        running_var *= momentum
        running_var += (1 - momentum) * var
        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                           END OF YOUR CODE                          #
        #######################################################################
    elif mode == 'test':
        #######################################################################
        # TODO: Implement the test-time forward pass for batch normalization. #
        # Use the running mean and variance to normalize the incoming data,   #
        # then scale and shift the normalized data using gamma and beta.      #
        # Store the result in the out variable.                               #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        # 实现 batch normalization 的 test-time 前向传播。
        pass
        # Using running mean and variance to normalize
        # 使用运行均值和方差normalize输入数据，
        std = np.sqrt(running_var + eps)
        xn = (x - running_mean) / std
        # 然后使用gamma和beta缩放和移位normalized数据。
        # 将结果存储在out变量中。
        out = gamma * xn + beta
        cache = (mode, x, xn, gamma, beta, std)
        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                          END OF YOUR CODE                           #
        #######################################################################
    else:
        raise ValueError('Invalid forward batchnorm mode "%s"' % mode)

    # Store the updated running means back into bn_param
    bn_param['running_mean'] = running_mean
    bn_param['running_var'] = running_var

    return out, cache


def batchnorm_backward(dout, cache):
    """
    Backward pass for batch normalization.

    For this implementation, you should write out a computation graph for
    batch normalization on paper and propagate gradients backward through
    intermediate nodes.
    对于这个实现，您应该在纸上写出一个batch normalization的计算图，并反向传播梯度，通过中间节点。
    Inputs:
    - dout: Upstream derivatives, of shape (N, D)
    - cache: Variable of intermediates from batchnorm_forward.

    Returns a tuple of:
    - dx: Gradient with respect to inputs x, of shape (N, D)
    - dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
    - dbeta: Gradient with respect to shift parameter beta, of shape (D,)
    """
    dx, dgamma, dbeta = None, None, None
    ###########################################################################
    # TODO: Implement the backward pass for batch normalization. Store the    #
    # results in the dx, dgamma, and dbeta variables.                         #
    # Referencing the original paper (https://arxiv.org/abs/1502.03167)       #
    # might prove to be helpful.                                              #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现batch normalization的反向传播。
    # 将结果存储在dx、dgamma和dbeta变量中。
#     https://zhuanlan.zhihu.com/p/21560667
    pass
    mode = cache[0]
    if mode == 'train':
        # 官方答案，见assignment3
#         mode, x, gamma, xc, std, xn, out = cache
        # 为了alternative也能适配，所以选择了这样的cache↓
        mode, x, gamma, mu, var, eps, xn = cache
        xc = x - mu
        std = np.sqrt(var + eps)
        
#     https://arxiv.org/abs/1502.03167
        N = x.shape[0]
        dbeta = np.sum(dout , axis = 0)
        dgamma = np.sum(xn * dout, axis=0)
        dxn = gamma * dout
        
        dxc = dxn / std
        dstd = -np.sum((dxn * xc) / (std * std), axis=0)
        dvar = 0.5 * dstd / std
        dxc += (2.0 / N) * xc * dvar
        dmu = np.sum(dxc, axis=0)
        dx = dxc - dmu / N
    elif mode == 'test':
        mode, x, xn, gamma, beta, std = cache
        dbeta = np.sum(dout , axis = 0)
        dgamma = np.sum(xn * dout, axis=0)
        dxn = gamma * dout
        dx = dxn / std
    else:
        raise ValueError(mode)

    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta


def batchnorm_backward_alt(dout, cache):
    """
    Alternative backward pass for batch normalization.
    可选的反向传播来batch normalization
    For this implementation you should work out the derivatives for the batch
    normalizaton backward pass on paper and simplify as much as possible. You
    should be able to derive a simple expression for the backward pass. 
    See the jupyter notebook for more hints.
    对于此实现，您应该在纸上计算出batch normalization反向传播的导数并尽可能地简化。
    您应该能够为反向传播推导出一个简单的表达式。
    Note: This implementation should expect to receive the same cache variable
    as batchnorm_backward, but might not use all of the values in the cache.
    此实现应该期望接收与batchnorm_backward相同的cache变量，但可能不会使用缓存中的所有值。
    Inputs / outputs: Same as batchnorm_backward
    """
    dx, dgamma, dbeta = None, None, None
    ###########################################################################
    # TODO: Implement the backward pass for batch normalization. Store the    #
    # results in the dx, dgamma, and dbeta variables.                         #
    #                                                                         #
    # After computing the gradient with respect to the centered inputs, you   #
    # should be able to compute gradients with respect to the inputs in a     #
    # single statement; our implementation fits on a single 80-character line.#
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现batch normalization的反向传播。
    # 将结果存储在dx、dgamma和dbeta变量中。
    # 在计算了相对于中心输入的梯度之后，您应该能够在一条语句中计算相对于输入的梯度;
    # 我们的实现只需要一行80个字符。
    # https://zhuanlan.zhihu.com/p/21560667
    # https://arxiv.org/abs/1502.03167
#     pass
    # Because no backprop in test mode
    mode, x, gamma, sample_mean, sample_var, eps, x_hat = cache
    M = x.shape[0]
    dgamma = np.sum(x_hat * dout, axis = 0)
    dbeta = np.sum(dout , axis = 0)
    dx_hat = dout * gamma
    dvar = np.sum(dx_hat * (x - sample_mean) * (-0.5) * np.power(sample_var + eps, -1.5), axis = 0)
    dmean = np.sum(dx_hat * -1 / np.sqrt(sample_var +eps), axis = 0) + dvar * np.mean(-2 * (x - sample_mean), axis =0)
    dx = 1 / np.sqrt(sample_var + eps) * dx_hat + dvar * 2.0 / M * (x-sample_mean) + 1.0 / M * dmean
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta


def layernorm_forward(x, gamma, beta, ln_param):
    """
    Forward pass for layer normalization.

    During both training and test-time, the incoming data is normalized per data-point,
    before being scaled by gamma and beta parameters identical to that of batch normalization.
    在训练和测试期间，输入数据被归一化每个数据点，然后使用与批量归一化相同的gamma和beta参数进行缩放。
    Note that in contrast to batch normalization, the behavior during train and test-time for
    layer normalization are identical, and we do not need to keep track of running averages
    of any sort.
    注意，与批量归一化不同，层归一化的训练和测试期间的行为是相同的，我们不需要跟踪任何类型的运行平均值。
    Input:
    - x: Data of shape (N, D)
    - gamma: Scale parameter of shape (D,)
    - beta: Shift paremeter of shape (D,)
    - ln_param: Dictionary with the following keys:
        - eps: Constant for numeric stability

    Returns a tuple of:
    - out: of shape (N, D)
    - cache: A tuple of values needed in the backward pass
    """
    out, cache = None, None
    eps = ln_param.get('eps', 1e-5)
    ###########################################################################
    # TODO: Implement the training-time forward pass for layer norm.          #
    # Normalize the incoming data, and scale and  shift the normalized data   #
    #  using gamma and beta.                                                  #
    # HINT: this can be done by slightly modifying your training-time         #
    # implementation of  batch normalization, and inserting a line or two of  #
    # well-placed code. In particular, can you think of any matrix            #
    # transformations you could perform, that would enable you to copy over   #
    # the batch norm code and leave it almost unchanged?                      #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现层归一化的训练时间前向传播。
    # 对传入数据进行归一化，并使用gamma和beta对归一化数据进行缩放和移位。
    # 这可以通过稍微修改批量归一化的训练时间实现，并插入一两行适当的代码来实现。
    # 特别是，您能想到您可以执行的任何矩阵变换，使您能够复制批量归一化代码并保持它几乎不变吗?
    pass
    x_T = x.T
    # Compute output
    # 使用minibatch统计数据来计算平均值和方差，
    # 使用这些统计数据来normalize输入数据，
    mu = x_T.mean(axis=0)
    xc_T = x_T - mu
    var = np.mean(xc_T ** 2, axis=0)
    std = np.sqrt(var + eps)
    xn_T = xc_T / std
    xn = xn_T.T
    # 使用gamma和beta来缩放和移动normalized数据。
    # 您应该将输出存储在变量out中。
    out = gamma * xn + beta
    # 用于反向传播的任何intermediates都应该存储在cache变量中。
    cache = (x, gamma, xc_T, std, xn_T, out) # 官方答案，见assignment3
#     cache = (x, xn, gamma, mu, var, eps)
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return out, cache


def layernorm_backward(dout, cache):
    """
    Backward pass for layer normalization.

    For this implementation, you can heavily rely on the work you've done already
    for batch normalization.
    对于此实现，您可以严重依赖已经完成的批量归一化工作。

    Inputs:
    - dout: Upstream derivatives, of shape (N, D)
    - cache: Variable of intermediates from layernorm_forward.

    Returns a tuple of:
    - dx: Gradient with respect to inputs x, of shape (N, D)
    - dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
    - dbeta: Gradient with respect to shift parameter beta, of shape (D,)
    """
    dx, dgamma, dbeta = None, None, None
    ###########################################################################
    # TODO: Implement the backward pass for layer norm.                       #
    #                                                                         #
    # HINT: this can be done by slightly modifying your training-time         #
    # implementation of batch normalization. The hints to the forward pass    #
    # still apply!                                                            #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现层归一化的反向传播
    # 这可以通过稍微修改批量归一化的训练时实现来实现。前向传播的提示仍然有效!
    pass
    x, gamma, xc_T, std, xn_T, out = cache # 官方答案，见assignment3
#     https://arxiv.org/abs/1502.03167
    x_T = x.T
    dout_T = dout.T
    
    N = x_T.shape[0]
    dbeta = dout.sum(axis=0)
    dgamma = np.sum(xn_T.T * dout, axis=0)
    dxn_T = dout_T * gamma[:,np.newaxis]

    dxc_T = dxn_T / std
    dstd = -np.sum((dxn_T * xc_T) / (std * std), axis=0)
    dvar = 0.5 * dstd / std
    dxc_T += (2.0 / N) * xc_T * dvar
    dmu = np.sum(dxc_T, axis=0)
    dx_T = dxc_T - dmu / N
    dx = dx_T.T
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx, dgamma, dbeta


def dropout_forward(x, dropout_param):
    """
    Performs the forward pass for (inverted) dropout.

    Inputs:
    - x: Input data, of any shape
    - dropout_param: A dictionary with the following keys:
      - p: Dropout parameter. We keep each neuron output with probability p.
      - mode: 'test' or 'train'. If the mode is train, then perform dropout;
        if the mode is test, then just return the input.
        如果模式是train，则执行dropout;如果模式是test，则返回输入。
      - seed: Seed for the random number generator. Passing seed makes this
        function deterministic, which is needed for gradient checking but not
        in real networks.
        种子为随机数生成器。传递种子使得该函数具有确定性，这是梯度检测所需要的，而在实际网络中则不需要。

    Outputs:
    - out: Array of the same shape as x.
    - cache: tuple (dropout_param, mask). In training mode, mask is the dropout
      mask that was used to multiply the input; in test mode, mask is None.

    NOTE: Please implement **inverted** dropout, not the vanilla version of dropout.
    请实现反向dropout，而不是普通版本的dropout。
    See http://cs231n.github.io/neural-networks-2/#reg for more details.

    NOTE 2: Keep in mind that p is the probability of **keep** a neuron
    output; this might be contrary to some sources, where it is referred to
    as the probability of dropping a neuron output.
    记住p是神经元输出的概率;这可能与某些来源相反，在那它被称为神经元输出dropping的概率。
    """
    p, mode = dropout_param['p'], dropout_param['mode']
    if 'seed' in dropout_param:
        np.random.seed(dropout_param['seed'])

    mask = None
    out = None

    if mode == 'train':
        #######################################################################
        # TODO: Implement training phase forward pass for inverted dropout.   #
        # Store the dropout mask in the mask variable.                        #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

        pass
        mask = (np.random.rand(*x.shape) < p) / p #<p 所以是keep prob
        out = x * mask
        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                           END OF YOUR CODE                          #
        #######################################################################
    elif mode == 'test':
        #######################################################################
        # TODO: Implement the test phase forward pass for inverted dropout.   #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

        pass
        out = x
        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                            END OF YOUR CODE                         #
        #######################################################################

    cache = (dropout_param, mask)
    out = out.astype(x.dtype, copy=False)

    return out, cache


def dropout_backward(dout, cache):
    """
    Perform the backward pass for (inverted) dropout.

    Inputs:
    - dout: Upstream derivatives, of any shape
    - cache: (dropout_param, mask) from dropout_forward.
    """
    dropout_param, mask = cache
    mode = dropout_param['mode']

    dx = None
    if mode == 'train':
        #######################################################################
        # TODO: Implement training phase backward pass for inverted dropout   #
        #######################################################################
        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

        pass
        dx = mask * dout
        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
        #######################################################################
        #                          END OF YOUR CODE                           #
        #######################################################################
    elif mode == 'test':
        dx = dout
    return dx


def conv_forward_naive(x, w, b, conv_param):
    """
    A naive implementation of the forward pass for a convolutional layer.
    卷积层的前向传播的简单实现。
    The input consists of N data points, each with C channels, height H and
    width W. We convolve each input with F different filters, where each filter
    spans all C channels and has height HH and width WW.
    输入由N个数据点组成，每个点有C个通道，高度H和宽度w。
    我们将每个输入与F个不同的滤波器进行卷积，每个滤波器spans所有C个通道，高度HH和宽度WW。
    
    Input:
    - x: Input data of shape (N, C, H, W)  (数量，通道个数，高，宽)
    - w: Filter weights of shape (F, C, HH, WW)  (filters个数，通道个数，高，宽)
    - b: Biases, of shape (F,)
    - conv_param: A dictionary with the following keys:
      - 'stride': The number of pixels between adjacent receptive fields in the
        horizontal and vertical directions.
        在水平和垂直方向上相邻接受域之间的像素数。
      - 'pad': The number of pixels that will be used to zero-pad the input. 
        将用于对输入进行零填充的像素数。
        

    During padding, 'pad' zeros should be placed symmetrically (i.e equally on both sides)
    along the height and width axes of the input. Be careful not to modfiy the original
    input x directly.
    在填充时，'pad'零应该对称放置(i.e两边相等)沿输入的高、宽轴。
    注意不要直接修改原始输入x。
    
    Returns a tuple of:(数量，filters个数，高，宽)
    - out: Output data, of shape (N, F, H', W') where H' and W' are given by
      H' = 1 + (H + 2 * pad - HH) / stride
      W' = 1 + (W + 2 * pad - WW) / stride
    - cache: (x, w, b, conv_param)
    """
    out = None
    ###########################################################################
    # TODO: Implement the convolutional forward pass.                         #
    # Hint: you can use the function np.pad for padding.                      #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # https://zhuanlan.zhihu.com/p/22038289
# 蓝色的输入数据 和 红色的滤波器 逐元素相乘→求其总和→加上偏差→输出的某个元素
    pass
    N, C, H, W = x.shape
    F, _, HH, WW = w.shape
    stride = conv_param['stride']
    pad = conv_param['pad']
    
#     '''
    # 初始化输出out (数量，filters个数，高，宽)
    H_out=int(1 + (H + 2 * pad - HH) / stride)
    W_out=int(1 + (W + 2 * pad - WW) / stride)
    out = np.zeros((N , F , H_out, W_out))
    
    # 在填充时，'pad'零应该对称放置(i.e两边相等)沿输入的高、宽轴。
    # 注意不要直接修改原始输入x。
    x_pad = np.pad(x, ((0,), (0,), (pad,), (pad,)), mode='constant', constant_values=0)
    
    # 通过遍历输出out的高和宽，对应F个输出的(i, j)位置
    for i in range(H_out):
        for j in range(W_out):
            # 在填充后的输入数据上，以stride步长抽取
            x_pad_masked = x_pad[:, :, i*stride:i*stride+HH, j*stride:j*stride+WW]
            # 遍历F个filters，第k个filter对应第k个输出的(i, j)位置
            for k in range(F):
                # 蓝色的输入数据 和 红色的滤波器 逐元素相乘→求其总和
                # (N, F, H_out,W_out) = (N, C, HH, WW)  * (F, C, HH, WW) 
                out[:, k , i, j] = np.sum(x_pad_masked * w[k, :, :, :], axis=(1,2,3))
            #out[:, : , i, j] = np.sum(x_pad_masked * w[:, :, :, :], axis=(1,2,3))

#     将b扩充为4维，对第k个输出，加上与其对应的偏置
#     for k in range(F):
#       out[:, k, :, :] = out[:, k, :, :] + b[k]
    out = out + (b)[None, :, None, None]
#     '''

    '''
    H_=int(1 + (H + 2 * pad - HH) / stride)
    W_=int(1 + (W + 2 * pad - WW) / stride)
    out=np.random.randn(x.shape[0],w.shape[0],H_,W_)
    x_pad=np.pad(x, ((0,0),(0,0),(1,1),(1,1)), 'constant', constant_values=((0,0),(0,0),(0,0),(0,0)))

    for ni in range(x.shape[0]):
        for fi in range(w.shape[0]):
            for xi in range(H_):
                for yi in range(W_):
                    out[ni, fi, xi, yi] = np.sum(x_pad[ni, :, xi * stride:xi * stride + HH, yi * stride:yi * stride + WW] * w[fi, :, :, :])

            out[ni,fi,:,:]+=b[fi]
#     '''
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    cache = (x, w, b, conv_param)
    return out, cache


def conv_backward_naive(dout, cache):
    """
    A naive implementation of the backward pass for a convolutional layer.

    Inputs:
    - dout: Upstream derivatives. (数量，filters个数，高，宽)
    - cache: A tuple of (x, w, b, conv_param) as in conv_forward_naive

    Returns a tuple of:
    - dx: Gradient with respect to x (数量，通道个数，高，宽)
    - dw: Gradient with respect to w (filters个数，通道个数，高，宽)
    - db: Gradient with respect to b
    """
    dx, dw, db = None, None, None
    ###########################################################################
    # TODO: Implement the convolutional backward pass.                        #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # https://zhuanlan.zhihu.com/p/22038289
    pass
    x, w, b, conv_param = cache

    N, C, H, W = x.shape
    F, _, HH, WW = w.shape
    stride = conv_param['stride']
    pad = conv_param['pad']
    H_out=int(1 + (H + 2 * pad - HH) / stride)
    W_out=int(1 + (W + 2 * pad - WW) / stride)

    x_pad = np.pad(x, ((0,), (0,), (pad,), (pad,)), mode='constant', constant_values=0)
    dx = np.zeros_like(x)
    dx_pad = np.zeros_like(x_pad)
    dw = np.zeros_like(w)
    db = np.zeros_like(b)
    db = np.sum(dout, axis = (0,2,3))

    for i in range(H_out):
        for j in range(W_out):
            x_pad_masked = x_pad[:, :, i*stride:i*stride+HH, j*stride:j*stride+WW]
            # out(N, F, H_out,W_out) = x_pad_masked(N, C, HH, WW)  * w(F, C, HH, WW)
            for k in range(F): #compute dw
                # dw(k,C,HH,WW) = x_pad_masked(N, C, HH, WW) * dout(N, F, H_out,W_out)
                dw[k ,: ,: ,:] += np.sum(x_pad_masked * (dout[:, k, i, j])[:, None, None, None], axis=0)
            for n in range(N): #compute dx_pad
                # dx_pad(N, C, HH, WW) = w(F, C, HH, WW) * dout(N, F, H_out,W_out)
                dx_pad[n, :, i*stride:i*stride+HH, j*stride:j*stride+WW] += np.sum((w[:, :, :, :] * (dout[n, :, i, j])[:,None ,None, None]), axis=0)
                
    dx = dx_pad[:,:,pad:-pad,pad:-pad]
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx, dw, db

def max_pool_forward_naive(x, pool_param):
    """
    A naive implementation of the forward pass for a max-pooling layer.

    Inputs:
    - x: Input data, of shape (N, C, H, W)
    - pool_param: dictionary with the following keys:
      - 'pool_height': The height of each pooling region
      - 'pool_width': The width of each pooling region
      - 'stride': The distance between adjacent pooling regions

    No padding is necessary here. Output size is given by 

    Returns a tuple of:
    - out: Output data, of shape (N, C, H', W') where H' and W' are given by
      H' = 1 + (H - pool_height) / stride
      W' = 1 + (W - pool_width) / stride
    - cache: (x, pool_param)
    """
    out = None
    ###########################################################################
    # TODO: Implement the max-pooling forward pass                            #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass
    N, C, H, W = x.shape
    HH = pool_param['pool_height']
    WW = pool_param['pool_width'] 
    stride = pool_param['stride']
    H_out = int((H-HH)/stride+1)
    W_out = int((W-WW)/stride+1)
    out = np.zeros((N,C,H_out,W_out))
    for i in range(H_out):
        for j in range(W_out):
            x_masked = x[:,:,i*stride : i*stride+HH, j*stride : j*stride+WW]
            out[:,:,i,j] = np.max(x_masked, axis=(2,3)) 
            
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    cache = (x, pool_param)
    return out, cache


def max_pool_backward_naive(dout, cache):
    """
    A naive implementation of the backward pass for a max-pooling layer.

    Inputs:
    - dout: Upstream derivatives
    - cache: A tuple of (x, pool_param) as in the forward pass.

    Returns:
    - dx: Gradient with respect to x
    """
    dx = None
    ###########################################################################
    # TODO: Implement the max-pooling backward pass                           #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****

    pass
    x, pool_param = cache
    N, C, H, W = x.shape
    HH = pool_param['pool_height']
    WW = pool_param['pool_width'] 
    stride = pool_param['stride']
    H_out = int((H-HH)/stride+1)
    W_out = int((W-WW)/stride+1)
    dx = np.zeros_like(x)

    for i in range(H_out):
        for j in range(W_out):
            x_masked = x[:,:,i*stride : i*stride+HH, j*stride : j*stride+WW]
            max_x_masked = np.max(x_masked,axis=(2,3))
            temp_binary_mask = (x_masked == (max_x_masked)[:,:,None,None])
            dx[:,:,i*stride : i*stride+HH, j*stride : j*stride+WW] += temp_binary_mask * (dout[:,:,i,j])[:,:,None,None]
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx


def spatial_batchnorm_forward(x, gamma, beta, bn_param):
    """
    Computes the forward pass for spatial batch normalization.

    Inputs:
    - x: Input data of shape (N, C, H, W)
    - gamma: Scale parameter, of shape (C,)
    - beta: Shift parameter, of shape (C,)
    - bn_param: Dictionary with the following keys:
      - mode: 'train' or 'test'; required
      - eps: Constant for numeric stability
      - momentum: Constant for running mean / variance. momentum=0 means that
        old information is discarded completely at every time step, while
        momentum=1 means that new information is never incorporated. The
        default of momentum=0.9 should work well in most situations.
        运行均值/方差的常数。
        momentum=0意味着在每一time步中，旧的信息会被完全丢弃，
        momentum=1意味着新信息永远不会被合并。
        momentum=0.9的默认值在大多数情况下都可以很好地工作。
      - running_mean: Array of shape (D,) giving running mean of features
      - running_var Array of shape (D,) giving running variance of features

    Returns a tuple of:
    - out: Output data, of shape (N, C, H, W)
    - cache: Values needed for the backward pass
    """
    out, cache = None, None

    ###########################################################################
    # TODO: Implement the forward pass for spatial batch normalization.       #
    #                                                                         #
    # HINT: You can implement spatial batch normalization by calling the      #
    # vanilla version of batch normalization you implemented above.           #
    # Your implementation should be very short; ours is less than five lines. #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现spatial batch normalization的前向传播。
    # 可以通过调用前面实现的batch normalization的普通版本来实现spatial batch normalization。
    # 你的实现应该很短;我们的少于五行。
    pass
    N, C, H, W = x.shape
    x_flat = x.transpose(0, 2, 3, 1).reshape(-1, C)
    out_flat, cache = batchnorm_forward(x_flat, gamma, beta, bn_param)
    out = out_flat.reshape(N, H, W, C).transpose(0, 3, 1, 2)
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return out, cache


def spatial_batchnorm_backward(dout, cache):
    """
    Computes the backward pass for spatial batch normalization.

    Inputs:
    - dout: Upstream derivatives, of shape (N, C, H, W)
    - cache: Values from the forward pass

    Returns a tuple of:
    - dx: Gradient with respect to inputs, of shape (N, C, H, W)
    - dgamma: Gradient with respect to scale parameter, of shape (C,)
    - dbeta: Gradient with respect to shift parameter, of shape (C,)
    """
    dx, dgamma, dbeta = None, None, None

    ###########################################################################
    # TODO: Implement the backward pass for spatial batch normalization.      #
    #                                                                         #
    # HINT: You can implement spatial batch normalization by calling the      #
    # vanilla version of batch normalization you implemented above.           #
    # Your implementation should be very short; ours is less than five lines. #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 可以通过调用前面实现的batch normalization的普通版本来实现spatial batch normalization。
    # 你的实现应该很短;我们的少于五行。
    pass
    N, C, H, W = dout.shape
    dout_flat = dout.transpose(0, 2, 3, 1).reshape(-1, C)
    dx_flat, dgamma, dbeta = batchnorm_backward(dout_flat, cache)
    dx = dx_flat.reshape(N, H, W, C).transpose(0, 3, 1, 2)
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################

    return dx, dgamma, dbeta


def spatial_groupnorm_forward(x, gamma, beta, G, gn_param):
    """
    Computes the forward pass for spatial group normalization.
    In contrast to layer normalization, group normalization splits each entry in the data into G contiguous pieces, which it then normalizes independently.
    Per feature shifting and scaling are then applied to the data, in a manner identical to that of batch normalization and layer normalization.
    计算空间组归一化的前向传播。
    与层归一化不同，组归一化将数据中的每个条目分割为G个连续的块，然后独立地对其进行归一化。
    然后对数据应用每个特征的移动和缩放，其方式与批量归一化和层归一化相同。
    Inputs:
    - x: Input data of shape (N, C, H, W)
    - gamma: Scale parameter, of shape (C,)
    - beta: Shift parameter, of shape (C,)
    - G: Integer mumber of groups to split into, should be a divisor of C
    要被分割成的组的整数数目，应该是C的除数
    - gn_param: Dictionary with the following keys:
      - eps: Constant for numeric stability

    Returns a tuple of:
    - out: Output data, of shape (N, C, H, W)
    - cache: Values needed for the backward pass
    """
    out, cache = None, None
    eps = gn_param.get('eps',1e-5)
    ###########################################################################
    # TODO: Implement the forward pass for spatial group normalization.       #
    # This will be extremely similar to the layer norm implementation.        #
    # In particular, think about how you could transform the matrix so that   #
    # the bulk of the code is similar to both train-time batch normalization  #
    # and layer normalization!                                                # 
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现空间组归一化的前向传播。
    # 这与层归一化实现非常相似。
    # 特别是，考虑如何转换矩阵，使代码的大部分类似于训练时批量归一化和层归一化!
    pass
   
    N, C, H, W = x.shape
    x_group = np.reshape(x, [N, G, C // G, H, W])  # 按G将C分组，注意需要确保C能被G整除，不然会报错
    mean = np.mean(x_group, axis=(2, 3, 4), keepdims=True)  # 均值
    var = np.var(x_group, axis=(2, 3, 4), keepdims=True)  # 方差
    x_group_norm = (x_group - mean) / np.sqrt(var + eps)  # 归一化
    x_norm = np.reshape(x_group_norm, [N, C, H, W])  # 还原维度
    out = gamma * x_norm + beta
    cache = (x_group, mean, var, x_norm, G, eps, gamma, beta)
    
    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return out, cache


def spatial_groupnorm_backward(dout, cache):
    """
    Computes the backward pass for spatial group normalization.

    Inputs:
    - dout: Upstream derivatives, of shape (N, C, H, W)
    - cache: Values from the forward pass

    Returns a tuple of:
    - dx: Gradient with respect to inputs, of shape (N, C, H, W)
    - dgamma: Gradient with respect to scale parameter, of shape (C,)
    - dbeta: Gradient with respect to shift parameter, of shape (C,)
    """
    dx, dgamma, dbeta = None, None, None

    ###########################################################################
    # TODO: Implement the backward pass for spatial group normalization.      #
    # This will be extremely similar to the layer norm implementation.        #
    ###########################################################################
    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    # 实现空间组归一化的反向传播。
    # 这与层归一化实现非常相似。
    pass

    x_group, mean, var, x_norm, G, eps, gamma, beta = cache
    N, C, H, W = dout.shape
    dbeta = np.sum(dout, axis=(0, 2, 3), keepdims=True)
    dgamma = np.sum(dout * x_norm, axis=(0, 2, 3), keepdims=True)

    # dx_group_norm
    dx_norm = dout * gamma
    dx_group_norm = np.reshape(dx_norm, [N, G, C // G, H, W])
    # dvar
    dvar = -0.5 * ((var + eps) ** (-1.5)) * np.sum(dx_group_norm * (x_group - mean), axis=(2, 3, 4), keepdims=True) 
    # dmean
    N_GROUP = C // G * H * W
    dmean1 = np.sum(dx_group_norm * -1.0 / np.sqrt(var + eps), axis=(2, 3, 4), keepdims=True)
    dmean2_var = dvar * -2.0 / N_GROUP * np.sum(x_group - mean, axis=(2, 3, 4), keepdims=True)
    dmean = dmean1 + dmean2_var
    # dx_group
    dx_group1 = dx_group_norm * 1.0 / np.sqrt(var + eps)
    dx_group_var = dvar * 2.0 / N_GROUP * (x_group - mean)
    dx_group_mean = dmean * 1.0 / N_GROUP
    dx_group = dx_group1 + dx_group_var + dx_group_mean
    # dx
    dx = np.reshape(dx_group, [N, C, H, W])

    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
    ###########################################################################
    #                             END OF YOUR CODE                            #
    ###########################################################################
    return dx, dgamma, dbeta


def svm_loss(x, y):
    """
    Computes the loss and gradient using for multiclass SVM classification.
    计算损失和梯度用于多类SVM分类。
    Inputs:
    - x: Input data, of shape (N, C) where x[i, j] is the score for the jth
      class for the ith input.
    - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and
      0 <= y[i] < C

    Returns a tuple of:
    - loss: Scalar giving the loss
    - dx: Gradient of the loss with respect to x
    """
    N = x.shape[0]
    correct_class_scores = x[np.arange(N), y]
    margins = np.maximum(0, x - correct_class_scores[:, np.newaxis] + 1.0)
    margins[np.arange(N), y] = 0
    loss = np.sum(margins) / N
    num_pos = np.sum(margins > 0, axis=1)
    dx = np.zeros_like(x)
    dx[margins > 0] = 1
    dx[np.arange(N), y] -= num_pos
    dx /= N
    return loss, dx


def softmax_loss(x, y):
    """
    Computes the loss and gradient for softmax classification.
    计算损失和梯度用于softmax分类。
    Inputs:
    - x: Input data, of shape (N, C) where x[i, j] is the score for the jth
      class for the ith input.
    - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and
      0 <= y[i] < C

    Returns a tuple of:
    - loss: Scalar giving the loss
    - dx: Gradient of the loss with respect to x
    """
    shifted_logits = x - np.max(x, axis=1, keepdims=True)
    Z = np.sum(np.exp(shifted_logits), axis=1, keepdims=True)
    log_probs = shifted_logits - np.log(Z)
    probs = np.exp(log_probs)
    N = x.shape[0]
    loss = -np.sum(log_probs[np.arange(N), y]) / N
    dx = probs.copy()
    dx[np.arange(N), y] -= 1
    dx /= N
    return loss, dx
